Mean energy classical

Figure 5.3. The classical picture the energies of dipoles varies continuously from parallel alignment with the applied magnetic field (-m-B) to antiparallel (+m-B). On the right is shown the distribution of molecules that results and the lower mean energy of the ensemble relative to the field-free environment.

To determine the optimal value of quantum correction y, several criteria have been proposed, all of which are based on the idea that an appropriate classical theory should correctly reproduce long-time hmits of the electronic populations. (Since the populations are proportional to the mean energy of the corresponding electronic oscillator, this condition also conserves the ZPE of this oscillator.) Employing phase-space theory, it has been shown that this requirement leads to the condition that the state-specihc level densities... 

Other analytical methods can also be applied for the detection of F in archaeological artefacts, especially when it is possible to take a sample or to perform microdestructive analysis. These are namely the electron microprobe with a wavelength-dispersive detector (WDX), secondary ion mass spectrometry (SIMS), X-ray fluorescence analysis under vacuum (XRF), transmission electron or scanning electron microscopy coupled with an energy-dispersive detector equipped with an ultrathin window (TEM/SEM-EDX). Fluorine can also be measured by means of classical potentiometry using an ion-selective electrode or ion chromatography. 

The most general vibrational motion of our solid is one in which each overtone vibrates simultaneously, with an arbitrary amplitude and phase. But in thermal equilibrium at temperature T, the various vibrations will be excited to quite definite extents. It proves to be mathematically the case that each of the overtones behaves just like an independent oscillator, whose frequency is the acoustical frequency of the overtone. Thus we can make immediate connections with the theory of the specific heats of oscillators, as we have done in Chap. XIII, Sec. 4. If the atoms vibrated according to the classical theory, then we should have equipartition, and at temperature T each oscillation would have the mean energy kT. This means that each of the N overtones would have equal... 

We denote by G the set of all the experimentally observable quantities (called physical observables) which must be reproduced. Such quantities are, for instance, the collision energy, the quantum numbers defining the intramolecular state (vibrations and the principal quantum number of rotation), the total angular momentum etc... However, there are other dynamical variables which have a clear meaning in Classical Mechanics but correspond to no physical observable because of the Uncertainty Principle. We call them phase variables and denote them globally by g. The phase variables must be given particular values to obtain, at given G, a particular trajectory. Such variables are, for instance, the various intramolecular normal vibrational phases, the intermolecular orientation, the secondary rotation quantum numbers, the impact parameter, etc... Thus we look for relationships of the type qo = qq (G, g) and either qo = qo (G, g) or po = Po (G, g)... 

The first two terms in the mean energy will be recognized as the familiar classical equipartition value of ksTf2 per degree of freedom. The remainder is due to the truncation of the atom-atom potential to have a finite dissociation energy. Due to this trimcation the contribution of the mean energy per bond does not continue to increase as the temperature increases ... 

The most obvious difficulty is that the numbers of oscillators in the classical RRK theory required to fit both the low pressure limit and the fall-off are smaller than the actual number of oscillators in the molecule, typically by a factor of about 2, although there can be wide variations [14]. Thus 14 oscillators have been used in the calculation presented in Fig. 5, but cyclobutane actually contains 30 oscillators. The problem is that the use of classical statistical mechanics overestimates the populations of excited states relative to the ground state. For example, in classical theory the mean energy of an oscillator is kT,... 

We shall limit ourselves to the case where the mean energy of each molecule is equal to the sum of the energies of translation, of rotation and of vibration. The heat capacity at constant volume (c/. 10.5) will also be composed of three terms arising from these three kinds of motion. The contribution from the translational motion is f R per mole, and that from rotation is JR or fR depending upon whether the molecule is linear or not. This last statement is only exact if the rotational motion may be treated by classical, as opposed to quantum, mechanics. This is a good approximation even at low temperatures except for very light molecules such as Hg and HD. Finally the contribution from vibration of the atoms in a molecule relative to one another is the sum of the contributions from the various modes of vibration. Each mode of vibration is characterized by a fundamental frequency vj which is independent of the temperature. It is convenient to relate the fundamental frequency to a characteristic temperature (0j) defined by... 

The power spectra of Ar3, shown in Fig. 2, make it clear that there is a distinct change between mean energies of —1.4 x 10-14 erg/atom (equivalent to 18.19 K) and —1.16 x 10 14 erg/atom (equivalent to 28.44 K). From sharp, distinct vibrations, the system has transformed to one with a continuous spectrum of available classical energies. 

A similar approach has been pursued by Daudey, Claverie and Malrieu, who used the same orthogonalization procedure for the orbitals but determined the interaction energy by means of classical perturbation theory. 

This result should be contrasted with expectations founded on the classical equipartition theorem which asserts that an oscillator should have a mean energy of kT. Here we note two ideas. First, in the low-temperature limit, the mean energy approaches a constant value that is just the zero point energy observed earlier in our quantum analysis of the same problem. Secondly, it is seen that in the high-temperature limit, the mean energy approaches the classical equipartition result E) = kT. This can be seen by expanding the exponential in the denominator of the second term to leading order in the inverse temperature (i.e. — 1 ... 

Einstein s explanation of these deviations is that it is not permissible in this case to use the classical expression for the mean energy of the oscillators, but that we must apply the expression obtained by Planck for the mean energy of a quantised oscillator. In that case the mean energy of the oscillators per mole is... 

In order to investigate the dynamics of chemical reactions by means of classical mechanical trajectory calculations it is desirable to have an analytic form for the potential energy surface so as to permit efficient calculation of the potential and its derivatives. All the empirical and most of the semi-empirical surfaces mentioned so far have been of this form, but all of them have been based on theoretical or physical models. In attempting either to find potential surfaces to describe specific reactions, or to investigate the effect of different features of a surface on the energy and angular distribution of the reaction products, several convenient and flexible functional forms for potential energy surfaces have been proposed. 

Experiment shows, however, that this is the case at high temperatures only, while, for low temperatures, c tends to zero. Einstein took Planck s value (5) for the mean energy instead of the classical one and obtained for one gram molecule ... 

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